two particles A and B start moving simultaneously along the line joining them in the same direction with acceleration of 1m/s^2 and 2m/s^2 and speed 3m/s and 1m/s respectively

initially A is 10m behind B

what is the minimum distance between them.

To solve the problem of two particles A and B moving along the same line with different accelerations and initial speeds, we need to analyze their motion mathematically. Let's break it down step by step.

Understanding the Initial Conditions

We have two particles:

• Particle A: Initial speed = 3 m/s, Acceleration = 1 m/s²
• Particle B: Initial speed = 1 m/s, Acceleration = 2 m/s²

Initially, particle A is 10 meters behind particle B. This means that if we consider the position of particle B as the reference point (0 meters), the initial position of particle A is at -10 meters.

Position Equations for Each Particle

The position of each particle as a function of time can be expressed using the kinematic equation:

s = ut + (1/2)at²

Where:

• s = displacement
• u = initial velocity
• a = acceleration
• t = time

Position of Particle A

For particle A:

• Initial velocity (u_A) = 3 m/s
• Acceleration (a_A) = 1 m/s²

The position of A as a function of time (t) is:

s_A = -10 + 3t + (1/2)(1)t²

Which simplifies to:

s_A = -10 + 3t + 0.5t²

Position of Particle B

For particle B:

• Initial velocity (u_B) = 1 m/s
• Acceleration (a_B) = 2 m/s²

The position of B as a function of time is:

s_B = 0 + 1t + (1/2)(2)t²

Which simplifies to:

s_B = t + t²

Finding the Distance Between the Particles

The distance between particles A and B at any time t can be expressed as:

d(t) = s_B - s_A

Substituting the expressions for s_A and s_B:

d(t) = (t + t²) - (-10 + 3t + 0.5t²)

This simplifies to:

d(t) = t + t² + 10 - 3t - 0.5t²

Combining like terms gives:

d(t) = 10 - 2t + 0.5t²

Finding the Minimum Distance

To find the minimum distance, we need to determine the critical points of the function d(t). We can do this by taking the derivative of d(t) and setting it to zero:

d'(t) = -2 + t

Setting the derivative equal to zero:

-2 + t = 0

Solving for t gives:

t = 2 seconds

Calculating the Minimum Distance

Now, we substitute t = 2 back into the distance equation:

d(2) = 10 - 2(2) + 0.5(2)²

Calculating this gives:

d(2) = 10 - 4 + 0.5(4)

d(2) = 10 - 4 + 2 = 8 meters

Final Thoughts

The minimum distance between particles A and B occurs at 2 seconds after they start moving, and that distance is 8 meters. This analysis shows how different accelerations and initial speeds can affect the relative motion of two objects. Understanding these concepts is crucial in physics, especially in kinematics.